In-Depth: Pythagoras Theorem and Applications
CLAT Application & Relevance
Importance: Medium. The Pythagoras Theorem is a fundamental concept in geometry that can be implicitly tested in CLAT QT, especially within caselets related to distances, dimensions of rectangular/square fields, or spatial arrangements. You might need to calculate a diagonal, a height, or a side length of a right-angled triangle formed within a larger figure (like a rectangle or square).
How it's tested: Finding the diagonal of a rectangular plot; calculating the height of a pole given its shadow and distance; problems involving distances in a coordinate plane (indirectly); finding missing dimensions in 2D or 3D figures that can be broken down into right triangles.
Section 1: Core Concept & Pythagoras Triplets
The Pythagoras Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
Key Formula
For a right-angled triangle with legs 'a' and 'b', and hypotenuse 'c':
a² + b² = c²
Converse of Pythagoras Theorem: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
Common Pythagorean Triplets (Memorize these for speed!)
These are sets of three positive integers a, b, and c, such that a² + b² = c². Recognizing these can save calculation time.
- (3, 4, 5) and its multiples (e.g., 6, 8, 10; 9, 12, 15; 30, 40, 50)
- (5, 12, 13) and its multiples (e.g., 10, 24, 26)
- (7, 24, 25)
- (8, 15, 17)
- (20, 21, 29)
Section 2: Solved CLAT-Style Examples (Application)
Example 1: Finding the Diagonal of a Rectangle (Caselet)
Passage Context: "A rectangular plot of land owned by a law university has a length of 24 meters and a width of 18 meters. A straight pathway is to be built diagonally across the plot to connect two opposite corners."
Question: "What will be the length of this diagonal pathway?"
Detailed Solution:
1. Visualize: The diagonal of a rectangle forms two right-angled triangles with the sides of the rectangle acting as the legs and the diagonal as the hypotenuse.
2. Identify Sides of the Right Triangle:
Leg a (width) = 18 meters
Leg b (length) = 24 meters
Hypotenuse c (diagonal) = ?
3. Apply Pythagoras Theorem: a² + b² = c²
18² + 24² = c²
324 + 576 = c²
900 = c²
4. Solve for c: c = √900 = 30 meters.
Alternatively (using triplets): Recognize (18, 24, ?) is a multiple of (3, 4, 5).
18 = 6 * 3
24 = 6 * 4
So, the hypotenuse will be 6 * 5 = 30.
Answer: The length of the diagonal pathway will be 30 meters.
Example 2: Combined Application with Areas (Caselet)
Passage Context: "A square-shaped courtroom has a side length of 20 meters. A triangular region within the courtroom, used for the witness box, has its base along one side of the courtroom wall, and its vertex is at the opposite corner of the courtroom. The height of this triangular region is 15 meters from the base."
Question: "What is the area of the witness box region?"
Detailed Solution:
1. Visualize the Triangle: The triangle has its base on one side of the square courtroom. Let's assume the base of the triangle is 'b' and its height is 'h'.
2. Identify Triangle Dimensions from Context:
The base of the triangle (b) is one side of the square = 20 meters.
The height of the triangle (h) from the base to the opposite corner = 15 meters.
3. Apply Area of Triangle Formula: Area = (1/2) * Base * Height
Area = (1/2) * 20 * 15
= 10 * 15 = 150 square meters.
Answer: The area of the witness box region is 150 square meters.
(Note: While Pythagoras Theorem is in the topic, this specific example primarily used area. However, a related question could ask for the distance from the vertex to a point on the base, forming a right-angled triangle where Pythagoras would be used.)